메뉴 건너뛰기




Volumn 5, Issue 1, 2007, Pages 1-51

Mathematical analysis of multi-scale models of complex fluids

Author keywords

Complex fluids; Doi Onsager equation; Micro macro models; Multiscale models; Numerical analysis; Well posedness

Indexed keywords


EID: 34248575233     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2007.v5.n1.a1     Document Type: Review
Times cited : (26)

References (90)
  • 2
    • 18844385711 scopus 로고    scopus 로고
    • Existence of global weak solution for some polymeric flow models
    • J. Barrett, C. Schwab and E. Süli, Existence of global weak solution for some polymeric flow models, Math. Models Methods Appl. Sci., 15, 939-983, 2005.
    • (2005) Math. Models Methods Appl. Sci , vol.15 , pp. 939-983
    • Barrett, J.1    Schwab, C.2    Süli, E.3
  • 3
    • 85039211258 scopus 로고    scopus 로고
    • J. Barrett and E. Süli, Existence of global weak solutions to kinetic models for dilute polymers, submitted to Multi. Model. Simul., 2006.
    • J. Barrett and E. Süli, Existence of global weak solutions to kinetic models for dilute polymers, submitted to Multi. Model. Simul., 2006.
  • 4
    • 0000815309 scopus 로고
    • Dynamics of Polymeric Liquids
    • 2nd edition, Wiley-Interscience, New York
    • R. B. Bird, O. Hassager, R. C. Armstrong and C. F. Curtiss, Dynamics of Polymeric Liquids; Vol. 1: Fluid Mechanics, 2nd edition, Wiley-Interscience, New York, 1987.
    • (1987) Fluid Mechanics , vol.1
    • Bird, R.B.1    Hassager, O.2    Armstrong, R.C.3    Curtiss, C.F.4
  • 5
    • 0000815309 scopus 로고
    • Dynamics of Polymeric Liquids
    • 2nd edition, Wiley-Interscience, New York
    • R. B. Bird, O. Hassager, R. C. Armstrong and C. F. Curtiss, Dynamics of Polymeric Liquids; Vol. 2: Kinetic Theory, 2nd edition, Wiley-Interscience, New York, 1987.
    • (1987) Kinetic Theory , vol.2
    • Bird, R.B.1    Hassager, O.2    Armstrong, R.C.3    Curtiss, C.F.4
  • 6
    • 0033567201 scopus 로고    scopus 로고
    • Variance reduction methods for CONFFESSIT-like simulations
    • J. Bovin and M. Picasso, Variance reduction methods for CONFFESSIT-like simulations, J. Non-Newtonian Fluid Mech., 84, 191-215, 1999.
    • (1999) J. Non-Newtonian Fluid Mech , vol.84 , pp. 191-215
    • Bovin, J.1    Picasso, M.2
  • 7
    • 33745419628 scopus 로고    scopus 로고
    • Nonlinear Fokker-Planck Navier-Stokes systems
    • P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems, Comm. Math. Sci., 3, 531-544, 2005.
    • (2005) Comm. Math. Sci , vol.3 , pp. 531-544
    • Constantin, P.1
  • 10
    • 11844304961 scopus 로고    scopus 로고
    • Note on the number of steady states for a 2D Smoluchowski equation
    • P. Constantin and J. Vukadinovic, Note on the number of steady states for a 2D Smoluchowski equation, Nonlinearity, 18, 441-443, 2005.
    • (2005) Nonlinearity , vol.18 , pp. 441-443
    • Constantin, P.1    Vukadinovic, J.2
  • 11
    • 33646692641 scopus 로고    scopus 로고
    • FENE dumbbell model and its several linear and nonlinear closure approximations
    • Q. Du, C. Liu and P. Yu, FENE dumbbell model and its several linear and nonlinear closure approximations, Multi. Model. Simul., 4, 709-731, 2005.
    • (2005) Multi. Model. Simul , vol.4 , pp. 709-731
    • Du, Q.1    Liu, C.2    Yu, P.3
  • 16
    • 85128805678 scopus 로고    scopus 로고
    • The heterogeneous multiscale methods
    • W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1, 87-132, 2003.
    • (2003) Comm. Math. Sci , vol.1 , pp. 87-132
    • E, W.1    Engquist, B.2
  • 17
    • 2942644114 scopus 로고    scopus 로고
    • Convergence of a stochastic method for the modeling of polymeric fluids, Acta Mathematicae Applicatae Sinica
    • W. E, T. Li and P. Zhang, Convergence of a stochastic method for the modeling of polymeric fluids, Acta Mathematicae Applicatae Sinica, English Series, 18, 529-536, 2002.
    • (2002) English Series , vol.18 , pp. 529-536
    • Li, W.E.T.1    Zhang, P.2
  • 18
    • 3242687156 scopus 로고    scopus 로고
    • Well-posedness of dumbbell model for the polymeric liquids
    • W. E, T. Li and P. Zhang, Well-posedness of dumbbell model for the polymeric liquids, Comm. Math. Phys., 248, 409-427, 2004.
    • (2004) Comm. Math. Phys , vol.248 , pp. 409-427
    • Li, W.E.T.1    Zhang, P.2
  • 19
    • 0022029535 scopus 로고
    • Viscosity, first normal stress coefficient, and molecular stretching in dilute polymer solutions
    • X. J. Fan, Viscosity, first normal stress coefficient, and molecular stretching in dilute polymer solutions, J. Non-Newtonian Fluid Mech., 17, 125-144, 1985.
    • (1985) J. Non-Newtonian Fluid Mech , vol.17 , pp. 125-144
    • Fan, X.J.1
  • 20
    • 0022046618 scopus 로고
    • Viscosity, first normal stress coefficient, and molecular stretching in concentrated solutions and melts
    • X. J. Fan, Viscosity, first normal stress coefficient, and molecular stretching in concentrated solutions and melts, J. Non-Newtonian Fluid Mech., 17, 251-265, 1985.
    • (1985) J. Non-Newtonian Fluid Mech , vol.17 , pp. 251-265
    • Fan, X.J.1
  • 21
    • 0000156539 scopus 로고    scopus 로고
    • The rigid-rod model for nematic polymers: An analysis of the shear flow problem
    • V. Faraoni, M. Grosso, S. Crescitelli and P. L. Maffettone, The rigid-rod model for nematic polymers: an analysis of the shear flow problem, J. Rheol., 43, 829-843, 1999.
    • (1999) J. Rheol , vol.43 , pp. 829-843
    • Faraoni, V.1    Grosso, M.2    Crescitelli, S.3    Maffettone, P.L.4
  • 22
    • 26944484957 scopus 로고    scopus 로고
    • A note on the Onsager model of nematic phase transitions
    • I. Fatkullin and V. Slastikov, A note on the Onsager model of nematic phase transitions, Comm. Math. Sci., 3, 21-26, 2005.
    • (2005) Comm. Math. Sci , vol.3 , pp. 21-26
    • Fatkullin, I.1    Slastikov, V.2
  • 23
    • 26944481099 scopus 로고    scopus 로고
    • Critical points of the Onsager functional on a sphere
    • I. Fatkullin and V. Slastikov, Critical points of the Onsager functional on a sphere, Nonlinearity, 18, 2565-2580, 2005.
    • (2005) Nonlinearity , vol.18 , pp. 2565-2580
    • Fatkullin, I.1    Slastikov, V.2
  • 28
    • 0037283001 scopus 로고    scopus 로고
    • Full-tensor alignment criteria for sheared nematic polymers
    • M. G. Forest, R. Zhou and Q. Wang, Full-tensor alignment criteria for sheared nematic polymers, J. Rheol., 47, 105-127, 2003.
    • (2003) J. Rheol , vol.47 , pp. 105-127
    • Forest, M.G.1    Zhou, R.2    Wang, Q.3
  • 29
    • 0000325280 scopus 로고
    • Existence results for the flow of viscoelastic fluids with a differnetial costitutive law
    • C. Guillope and J. C. Saut, Existence results for the flow of viscoelastic fluids with a differnetial costitutive law, Nonlinear Anal., 15, 849-869, 1990.
    • (1990) Nonlinear Anal , vol.15 , pp. 849-869
    • Guillope, C.1    Saut, J.C.2
  • 30
    • 0032215863 scopus 로고    scopus 로고
    • The Lagrangian particle method for macroscopic and micro-macro viscoelastic flow computations
    • P. Halin, G. Lielens, R. Keunings and V. Legat, The Lagrangian particle method for macroscopic and micro-macro viscoelastic flow computations, J. Non-Newtonian Fluid Mech., 79, 387-403, 1998.
    • (1998) J. Non-Newtonian Fluid Mech , vol.79 , pp. 387-403
    • Halin, P.1    Lielens, G.2    Keunings, R.3    Legat, V.4
  • 32
    • 0010534606 scopus 로고    scopus 로고
    • American Mathematical Society, Providence, Rhode Island
    • E. P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Society, Providence, Rhode Island, 2002.
    • (2002) Stochastic Analysis on Manifolds
    • Hsu, E.P.1
  • 33
    • 0032343437 scopus 로고    scopus 로고
    • The variational formulation of the Fokker-Planck equation
    • R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29, 1-17, 1998.
    • (1998) SIAM J. Math. Anal , vol.29 , pp. 1-17
    • Jordan, R.1    Kinderlehrer, D.2    Otto, F.3
  • 35
    • 7744234690 scopus 로고    scopus 로고
    • On a variance reduction technique for the micro-macro simulations of polymeric fluids
    • B. Jourdain, C. Le Bris and T. Lelièvre, On a variance reduction technique for the micro-macro simulations of polymeric fluids, J. Non-Newtonian Fluid Mech., 122, 91-106, 2004.
    • (2004) J. Non-Newtonian Fluid Mech , vol.122 , pp. 91-106
    • Jourdain, B.1    Le Bris, C.2    Lelièvre, T.3
  • 36
    • 33747504035 scopus 로고    scopus 로고
    • B. Jourdain, C. Le Bris and T. Lelièvre, Coupling PDEs and SDEs: the illustrative example of the multiscale simulation of viscoelasic flows, in Multiscale Methods in Science and Engineering, B. Engquist, P. Lötstedt, O. Runborg (Eds.), CSE, Springer, 44, 151-170, 2005.
    • B. Jourdain, C. Le Bris and T. Lelièvre, Coupling PDEs and SDEs: the illustrative example of the multiscale simulation of viscoelasic flows, in Multiscale Methods in Science and Engineering, B. Engquist, P. Lötstedt, O. Runborg (Eds.), CSE, Springer, 44, 151-170, 2005.
  • 37
    • 0036710134 scopus 로고    scopus 로고
    • Numerical analysis of micro-macro simulations of polymeric fluid flows: A simple case
    • B. Jourdain, T. Lelièvre and C. Le Bris, Numerical analysis of micro-macro simulations of polymeric fluid flows: a simple case, Math. Models Methods Appl. Sci., 12, 1205-1243, 2002.
    • (2002) Math. Models Methods Appl. Sci , vol.12 , pp. 1205-1243
    • Jourdain, B.1    Lelièvre, T.2    Le Bris, C.3
  • 38
    • 12144286593 scopus 로고    scopus 로고
    • Existence of solution for a micro-macro model of polymeric fluid: The FENE model
    • B. Jourdain, T. Lelièvre and C. Le Bris, Existence of solution for a micro-macro model of polymeric fluid: the FENE model, J. Funct. Anal., 209, 162-193, 2004.
    • (2004) J. Funct. Anal , vol.209 , pp. 162-193
    • Jourdain, B.1    Lelièvre, T.2    Le Bris, C.3
  • 41
    • 85039189125 scopus 로고    scopus 로고
    • R. Keunings, Finite element methods for integral viscoelastic fluids, in Rheology Reviews 2003, D. M. Binding and K. Walters (Eds.), British Society of Rheology, Aberystwyth, Wales, 167-195, 2003.
    • R. Keunings, Finite element methods for integral viscoelastic fluids, in Rheology Reviews 2003, D. M. Binding and K. Walters (Eds.), British Society of Rheology, Aberystwyth, Wales, 167-195, 2003.
  • 42
    • 85039204855 scopus 로고    scopus 로고
    • R. Keunings, Micro-macro methods for the multiscale simulation of viscoelastic flow using molecular models of kenitic theory, in Rheology Reviews 2004, 67-98, D. M. Binding and K. Walters (Eds.), British Society of Rheology, Aberystwyth, Wales, 2004.
    • R. Keunings, Micro-macro methods for the multiscale simulation of viscoelastic flow using molecular models of kenitic theory, in Rheology Reviews 2004, 67-98, D. M. Binding and K. Walters (Eds.), British Society of Rheology, Aberystwyth, Wales, 2004.
  • 45
    • 0027608224 scopus 로고
    • Calculation of viscoelastic flow using molecular models: The CONNFFESSIT approach
    • M. Laso and H. C. Öttinger, Calculation of viscoelastic flow using molecular models: the CONNFFESSIT approach, J. Non-Newtonian Fluid Mech., 47, 1-20, 1993.
    • (1993) J. Non-Newtonian Fluid Mech , vol.47 , pp. 1-20
    • Laso, M.1    Öttinger, H.C.2
  • 47
    • 4243180535 scopus 로고    scopus 로고
    • Stochastic models of polymeric fluids at small Deborah number
    • T. Li, E. Vanden-Eijnden, P. Zhang and W. E, Stochastic models of polymeric fluids at small Deborah number, J. Non-Newtonian Fluid Mech., 121, 117-125, 2004.
    • (2004) J. Non-Newtonian Fluid Mech , vol.121 , pp. 117-125
    • Li, T.1    Vanden-Eijnden, E.2    Zhang, P.3    E, W.4
  • 48
    • 2942679256 scopus 로고    scopus 로고
    • Local existence for the dumbbell model of polymeric fluids
    • T. Li, H. Zhang and P. Zhang, Local existence for the dumbbell model of polymeric fluids, Comm. PDE, 29, 903-923, 2004.
    • (2004) Comm. PDE , vol.29 , pp. 903-923
    • Li, T.1    Zhang, H.2    Zhang, P.3
  • 49
    • 33751523664 scopus 로고    scopus 로고
    • Convergence analysis of the BCF methods for high dimensional Hookean dumbbell model
    • T. Li and P. Zhang, Convergence analysis of the BCF methods for high dimensional Hookean dumbbell model, Multi. Model. Simul., 5, 205-234, 2006.
    • (2006) Multi. Model. Simul , vol.5 , pp. 205-234
    • Li, T.1    Zhang, P.2
  • 50
    • 34248556443 scopus 로고    scopus 로고
    • Analysis of 1+1 dimensional stochastic models of liquid crystal polymer flows
    • T. Li, P. Zhang and X. Zhou, Analysis of 1+1 dimensional stochastic models of liquid crystal polymer flows, Comm. Math. Sci., 2, 295-316, 2004.
    • (2004) Comm. Math. Sci , vol.2 , pp. 295-316
    • Li, T.1    Zhang, P.2    Zhou, X.3
  • 51
    • 0034376947 scopus 로고    scopus 로고
    • Existence of solutions for the Ericksen-Leslie system
    • F. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rat. Mech. Anal., 154, 135-156, 2000.
    • (2000) Arch. Rat. Mech. Anal , vol.154 , pp. 135-156
    • Lin, F.1    Liu, C.2
  • 52
    • 27844607354 scopus 로고    scopus 로고
    • On hydrodynamics of viscoelastic fluids
    • F. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58, 1437-1471, 2005.
    • (2005) Comm. Pure Appl. Math , vol.58 , pp. 1437-1471
    • Lin, F.1    Liu, C.2    Zhang, P.3
  • 54
    • 0034381125 scopus 로고    scopus 로고
    • Global solutions for some Oldroyd models of non-Newtonian flows
    • P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. of Math., 21B, 131-146, 2000.
    • (2000) Chin. Ann. of Math , vol.21 B , pp. 131-146
    • Lions, P.L.1    Masmoudi, N.2
  • 55
    • 34248519590 scopus 로고    scopus 로고
    • Boundary conditions for the microscopic FENE models
    • submitted
    • C. Liu and H. Liu, Boundary conditions for the microscopic FENE models, SIAM J. Appl. Math., submitted, 2006.
    • (2006) SIAM J. Appl. Math
    • Liu, C.1    Liu, H.2
  • 56
    • 26944448390 scopus 로고    scopus 로고
    • Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential
    • H. Liu, H. Zhang and P. Zhang, Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential, Comm. Math. Sci., 3, 201-218, 2005.
    • (2005) Comm. Math. Sci , vol.3 , pp. 201-218
    • Liu, H.1    Zhang, H.2    Zhang, P.3
  • 57
    • 84968499110 scopus 로고
    • Convergence of the random vortex method in two dimensions
    • D.-G. Long, Convergence of the random vortex method in two dimensions, J. Amer. Math. Soc., 1, 779-804, 1988.
    • (1988) J. Amer. Math. Soc , vol.1 , pp. 779-804
    • Long, D.-G.1
  • 58
    • 0042657876 scopus 로고    scopus 로고
    • A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model
    • A. Lozinski and C. Chauviere, A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model, J. Comp. Phys., 189, 607-625, 2003.
    • (2003) J. Comp. Phys , vol.189 , pp. 607-625
    • Lozinski, A.1    Chauviere, C.2
  • 59
    • 0038680697 scopus 로고    scopus 로고
    • A Fokker-Planck simulation of fast flows of melts and concentracted polymer solutions in complex solutions in complex geometries
    • A. Lozinski, C. Chauviere, J. Fan and R. G. Owens, A Fokker-Planck simulation of fast flows of melts and concentracted polymer solutions in complex solutions in complex geometries, J. Rheol., 47, 535-561, 2003.
    • (2003) J. Rheol , vol.47 , pp. 535-561
    • Lozinski, A.1    Chauviere, C.2    Fan, J.3    Owens, R.G.4
  • 60
    • 11844283929 scopus 로고    scopus 로고
    • The structure of equilibrium solutions of the one-dimensional Doi equation
    • C. Luo, H. Zhang and P. W. Zhang, The structure of equilibrium solutions of the one-dimensional Doi equation, Nonlinearity, 18, 379-389, 2005.
    • (2005) Nonlinearity , vol.18 , pp. 379-389
    • Luo, C.1    Zhang, H.2    Zhang, P.W.3
  • 61
    • 0029347345 scopus 로고
    • Bifurcation analysis of a molecular model for nematic polymers in shear flows
    • P. L. Maffettone and S. Crescitelli, Bifurcation analysis of a molecular model for nematic polymers in shear flows, J. Non-Newtonian Fluid Mech., 59, 73-91, 1995.
    • (1995) J. Non-Newtonian Fluid Mech , vol.59 , pp. 73-91
    • Maffettone, P.L.1    Crescitelli, S.2
  • 62
    • 0001355304 scopus 로고
    • A simple molecular-statistics theory of the nematic liquid-crystalline state
    • W. Maier and A. Z. Saupe, A simple molecular-statistics theory of the nematic liquid-crystalline state, Z. Naturforsch, 13a, 564-570, 1958;
    • (1958) Z. Naturforsch , vol.13 a , pp. 564-570
    • Maier, W.1    Saupe, A.Z.2
  • 65
    • 0024755190 scopus 로고
    • Description of the liquid crystalline phase of rodlike polymers at high shear rates
    • G. Marrucci and P. L. Maffettone, Description of the liquid crystalline phase of rodlike polymers at high shear rates, Macromolecules, 22, 4076-4082, 1989.
    • (1989) Macromolecules , vol.22 , pp. 4076-4082
    • Marrucci, G.1    Maffettone, P.L.2
  • 66
    • 85039238052 scopus 로고    scopus 로고
    • H. P. McKean, Propagation of chaos for a class of nonlinear parabolic equations, Lecture series in differential equations, Catholic University, Washington D.C., 7, 41-57, 1967.
    • H. P. McKean, Propagation of chaos for a class of nonlinear parabolic equations, Lecture series in differential equations, Catholic University, Washington D.C., 7, 41-57, 1967.
  • 67
    • 0000258933 scopus 로고
    • Variance reduced simulations of stochastic differential equations
    • M. Melchior and H. C. Öttinger, Variance reduced simulations of stochastic differential equations, J. Chem. Phys., 103, 9506-9509, 1995.
    • (1995) J. Chem. Phys , vol.103 , pp. 9506-9509
    • Melchior, M.1    Öttinger, H.C.2
  • 68
    • 0000624731 scopus 로고    scopus 로고
    • Variance reduced simulations of polymer dynamics
    • M. Melchior and H. C. Öttinger, Variance reduced simulations of polymer dynamics, J. Chem. Phys., 105, 3316-3331, 1996.
    • (1996) J. Chem. Phys , vol.105 , pp. 3316-3331
    • Melchior, M.1    Öttinger, H.C.2
  • 70
    • 0040554658 scopus 로고
    • A martingale approach to the Law of Large Numbers for weakly interacting stochastic processes
    • B. Oelschlager, A martingale approach to the Law of Large Numbers for weakly interacting stochastic processes, Ann. Prob., 12, 458-479, 1984.
    • (1984) Ann. Prob , vol.12 , pp. 458-479
    • Oelschlager, B.1
  • 72
    • 84971301149 scopus 로고
    • The effects of shape on the interaction of colloidal particles
    • L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N.Y. Acad. Sci., 51, 627-659, 1949.
    • (1949) Ann. N.Y. Acad. Sci , vol.51 , pp. 627-659
    • Onsager, L.1
  • 73
  • 77
    • 0346957816 scopus 로고
    • Local existence of solutions of the dirichlet initial-boundary value problem for incompressible hypoelastic materials
    • M. Renardy, Local existence of solutions of the dirichlet initial-boundary value problem for incompressible hypoelastic materials, SIAM J. Math. Anal., 21, 1369-1385, 1990.
    • (1990) SIAM J. Math. Anal , vol.21 , pp. 1369-1385
    • Renardy, M.1
  • 78
    • 0005228558 scopus 로고
    • An existence theorem for model equations resulting from kinetic theories of polymer solutions
    • M. Renardy, An existence theorem for model equations resulting from kinetic theories of polymer solutions, SIAM J. Math. Anal., 22, 313-327, 1991.
    • (1991) SIAM J. Math. Anal , vol.22 , pp. 313-327
    • Renardy, M.1
  • 79
    • 0001161978 scopus 로고    scopus 로고
    • Self-consistent-field theories for complex fluids
    • F. Schmid, Self-consistent-field theories for complex fluids, J. Phys.: Condens. Matter, 10, 8105-8138, 1998.
    • (1998) J. Phys.: Condens. Matter , vol.10 , pp. 8105-8138
    • Schmid, F.1
  • 80
    • 0036434051 scopus 로고    scopus 로고
    • Molecular orientation effects in viscoelasticity
    • J. K. C. Suen, Y. L. Joo and R. C. Armstrong, Molecular orientation effects in viscoelasticity, Ann. Rev. Fluid Mech., 34, 417-444, 2002.
    • (2002) Ann. Rev. Fluid Mech , vol.34 , pp. 417-444
    • Suen, J.K.C.1    Joo, Y.L.2    Armstrong, R.C.3
  • 81
    • 34248534649 scopus 로고
    • S. Albeverio ed, Pitman Research Notes in Mathematics
    • A. S. Sznitman, Topics in propogation of chaos, S. Albeverio ed., Pitman Research Notes in Mathematics, 124, 167-251, 1985.
    • (1985) Topics in propogation of chaos , vol.124 , pp. 167-251
    • Sznitman, A.S.1
  • 82
    • 0000336436 scopus 로고    scopus 로고
    • Viscoelastic phase separation
    • H. Tanaka, Viscoelastic phase separation, J. Phys.: Condens. Matter, 12, 207-264, 2000.
    • (2000) J. Phys.: Condens. Matter , vol.12 , pp. 207-264
    • Tanaka, H.1
  • 83
    • 0347638498 scopus 로고
    • Kinetic theory and rheology of dilute suspensions of finitely extensible dumbbells
    • H. R. Warner, Kinetic theory and rheology of dilute suspensions of finitely extensible dumbbells, Ind. Eng. Chem. Fundam., 11, 379-387, 1972.
    • (1972) Ind. Eng. Chem. Fundam , vol.11 , pp. 379-387
    • Warner, H.R.1
  • 84
    • 0037157328 scopus 로고    scopus 로고
    • A hydrodynamic theory for solutions of nonhomogeneous nematic liquid crystalline polymers of different configurations
    • Qi Wang, A hydrodynamic theory for solutions of nonhomogeneous nematic liquid crystalline polymers of different configurations, J. Chem. Phys., 116, 9120-9136, 2002.
    • (2002) J. Chem. Phys , vol.116 , pp. 9120-9136
    • Wang, Q.1
  • 85
    • 29244436403 scopus 로고    scopus 로고
    • From micro to macro dynamics via a new closure approximation to the FENE model of polymeric fluids
    • P. Yu, Q. Du and C. Liu, From micro to macro dynamics via a new closure approximation to the FENE model of polymeric fluids, Multi. Model. Simul., 3, 895-917, 2005.
    • (2005) Multi. Model. Simul , vol.3 , pp. 895-917
    • Yu, P.1    Du, Q.2    Liu, C.3
  • 86
    • 3042712390 scopus 로고    scopus 로고
    • A theoretical and numerical study for the rod-like model of polymeric fluid
    • H. Zhang and P. Zhang, A theoretical and numerical study for the rod-like model of polymeric fluid, J. Comp. Math., 22, 319-330, 2004.
    • (2004) J. Comp. Math , vol.22 , pp. 319-330
    • Zhang, H.1    Zhang, P.2
  • 87
    • 33646894129 scopus 로고    scopus 로고
    • Local existence for the FENE dumbbell model of polymeric fluids
    • H. Zhang and P. Zhang, Local existence for the FENE dumbbell model of polymeric fluids, Arch. Rat. Mech. Anal., 181, 373-400, 2006.
    • (2006) Arch. Rat. Mech. Anal , vol.181 , pp. 373-400
    • Zhang, H.1    Zhang, P.2
  • 89
    • 18344362303 scopus 로고    scopus 로고
    • Bending and base-stacking interactions in double-stranded DNA
    • H. Zhou, Y. Zhang and Z. Ouyang, Bending and base-stacking interactions in double-stranded DNA, Phys. Rev. Lett., 82, 4560-4563, 1999.
    • (1999) Phys. Rev. Lett , vol.82 , pp. 4560-4563
    • Zhou, H.1    Zhang, Y.2    Ouyang, Z.3
  • 90
    • 26944494507 scopus 로고    scopus 로고
    • A new proof on axisymmetric equilibria of a three-dimensional Smoluchowski equation
    • H. Zhou, H. Wang, M. Forest and Q. Wang, A new proof on axisymmetric equilibria of a three-dimensional Smoluchowski equation, Nonlinearity, 18, 2815-2825, 2005.
    • (2005) Nonlinearity , vol.18 , pp. 2815-2825
    • Zhou, H.1    Wang, H.2    Forest, M.3    Wang, Q.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.